--- jupytext: formats: md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.16.4 kernelspec: display_name: Python 3 language: python name: python3 --- (jax-301-refs)= # Autodiff with refs Refs (JAX's mutable arrays, introduced in {ref}`jax-101-refs`) interact with automatic differentiation in ways that immutable values can't express: plumbing data out of backward passes, accumulating gradients in place across many contributions, and differentiating with respect to mutable state itself. This page covers the interaction in detail, ending with a recipe that uses refs to relax `jax.lax.scan`'s fixed access pattern. ```{code-cell} import jax import jax.numpy as jnp ``` ## `Ref`s and automatic differentiation Autodiff can be applied to pure functions as before, even if they use array refs internally. For example: ```{code-cell} @jax.jit def pure(x): ref = jax.new_ref(x) ref[...] = ref[...] + ref[...] return ref[...] print(jax.grad(pure)(3.0)) # 2.0 ``` Autodiff can also be applied to functions that take array refs as arguments. The simplest case is when those ref arguments are only used for plumbing, and aren't involved in differentiation. For example, `jax.grad` differentiates with respect to its function's first argument by default, so ref arguments in other positions just pass through. Only non-differentiated values can be written into such plumbing refs: ```{code-cell} # error under jax.grad: writes a differentiated value into a plumbing ref def bad(x, some_plumbing_ref): y = x + x some_plumbing_ref[...] += y return y # fine def good(x, some_plumbing_ref): y = x + x some_plumbing_ref[...] += jax.lax.stop_gradient(y) return y ``` Differentiating _with respect to_ a ref-typed argument is another matter: pointing `jax.grad` at one (e.g. via `argnums`) is an error, since the gradient for a ref must itself live in a ref. Instead, use `jax.vjp` and `with_refs`, described below. You can combine plumbing refs with {func}`~jax.custom_vjp` to plumb data out of the backward pass of a differentiated function. (For a purely functional, drop-by-default alternative, see backward-pass logging with hijax primitives, {ref}`jax-301-bwd-logging`.) ```{code-cell} # First, define the helper `stash_grads`: @jax.custom_vjp def stash_grads(grads_ref, x): return x def stash_grads_fwd(grads_ref, x): return x, grads_ref def stash_grads_bwd(grads_ref, g): grads_ref[...] = g return None, g stash_grads.defvjp(stash_grads_fwd, stash_grads_bwd) ``` ```{code-cell} # Now, use `stash_grads` to stash intermediate gradients: def f(x, grads_ref): x = stash_grads(grads_ref, x) x = jnp.sin(x) return x grads_ref = jax.new_ref(0.) jax.grad(f)(1., grads_ref) print(grads_ref) # Ref(0.54), the gradient at the stash point: cos(1.) ``` Notice that `stash_grads_fwd` returns a `Ref` here. That's a special allowance for `custom_vjp` fwd rules: it's really syntax for indicating which ref arguments the fwd and bwd rules share. So any refs returned by a fwd rule must be arguments to that fwd rule. ## Differentiating with respect to `Ref` arguments The plumbing refs above just carry data out of the computation, but no gradients flow through them. We can also differentiate _with respect to_ a ref argument. Since the gradient for a `Ref`-typed input is itself `Ref`-typed, `jax.grad` doesn't apply here. Instead we use `jax.vjp`, and bind a gradient ref to the VJP function using its `with_refs` method: ```{code-cell} def f(x_ref): return x_ref[...] ** 2 x_ref = jax.new_ref(2.) y, f_vjp = jax.vjp(f, x_ref) x_grad_ref = jax.new_ref(0.) f_vjp.with_refs(x_grad_ref)(1.0) # bind the gradient ref, then apply the VJP print(x_grad_ref) # Ref(4.) ``` Here `with_refs` takes one entry per argument of the differentiated function and returns a new VJP function with those gradient refs bound. When differentiating with respect to a ref argument, using `with_refs` is mandatory; the gradient needs a ref to be accumulated into, so calling the VJP function without binding one is an error: ```{code-cell} _, f_vjp = jax.vjp(f, jax.new_ref(2.)) try: f_vjp(1.0) # error! no ref bound for the ref-typed argument's gradient except Exception as e: print(e) # ... gradient must be accumulated into a ref ... `with_refs` ... ``` The gradient is accumulated into the bound ref via `+=`, added to whatever the ref already contains rather than overwriting it: ```{code-cell} x_grad_ref = jax.new_ref(100.) _, f_vjp = jax.vjp(f, jax.new_ref(2.)) f_vjp.with_refs(x_grad_ref)(1.0) print(x_grad_ref) # Ref(104.), i.e. 100. + 4.: accumulated, not set ``` Accumulating rather than setting might seem like an odd choice, but it means one gradient ref can collect contributions from several backward passes with no extra memory traffic, as in the examples below. In-place updates to the ref argument inside the differentiated function are differentiated too. The result is the gradient with respect to the ref's initial value: ```{code-cell} def g(x_ref): x_ref[...] = jnp.sin(x_ref[...]) return x_ref[...] ** 2 x_ref = jax.new_ref(2.) _, g_vjp = jax.vjp(g, x_ref) # runs g, so x_ref is updated in place here g_grad_ref = jax.new_ref(0.) g_vjp.with_refs(g_grad_ref)(1.0) print(g_grad_ref) # Ref(-0.757), i.e. 2*sin(2)*cos(2) ``` ## Gradient refs for value arguments When differentiating with respect to an ordinary `Array` argument, `with_refs` is optional: we can call the VJP function directly and get the gradient back as a value in the usual way, or we can bind a ref and have the gradient accumulated into it in place: ```{code-cell} _, sin_vjp = jax.vjp(jnp.sin, 1.0) x_bar, = sin_vjp(1.0) # the usual way: gradient returned as a value print(x_bar) # 0.54 grad_ref = jax.new_ref(0.) _, sin_vjp = jax.vjp(jnp.sin, 1.0) sin_vjp.with_refs(grad_ref)(1.0) # gradient accumulated into grad_ref print(grad_ref) # Ref(0.54) ``` We can mix and match. Each entry of `with_refs` can be: * a `Ref`, meaning accumulate this argument's gradient into the ref in place (the VJP call then returns a `jax.ad.GradRef()` placeholder in that position); * `jax.ad.GradValue()`, meaning return this argument's gradient as a value in the usual way (the default); or * `jax.ad.DontWant()`, meaning don't compute this argument's gradient at all (the VJP call returns a `jax.ad.DidntWant()` placeholder in that position; more on this below). One reason to use a gradient ref here is to exploit sparsity. Consider differentiating a function that slices its input: ```{code-cell} @jax.jit def take(x, i): return x[i] x = jnp.arange(10.) _, take_vjp = jax.vjp(take, x, 3) x_bar, _ = take_vjp(1.0) print(x_bar) # [0., 0., 0., 1., 0., 0., 0., 0., 0., 0.] ``` The gradient with respect to `x` is one-hot: the backward pass materializes a dense array of zeros and sets a single element of it. If we compute many such gradients and sum them, say over a loop of sparse accesses, we pay for a dense array's worth of memory traffic on each one, even though each contribution only touches one element. If we instead bind a gradient ref with `with_refs`, each backward pass performs a sparse in-place add-update, writing only where it needs to: ```{code-cell} grad_ref = jax.new_ref(jnp.zeros(10)) for i in [3, 5, 3]: _, take_vjp = jax.vjp(take, x, i) take_vjp.with_refs(grad_ref, jax.ad.GradValue())(1.0) # no gradient ref for i print(grad_ref) # Ref([0., 0., 0., 2., 0., 1., 0., 0., 0., 0.]) ``` We can check that the update is sparse by inspecting the jaxpr of a single VJP application: ```{code-cell} @jax.make_jaxpr def take_vjp_jaxpr(): _, take_vjp = jax.vjp(take, x, 3) take_vjp.with_refs(grad_ref, jax.ad.GradValue())(1.0) print(take_vjp_jaxpr()) ``` The backward pass boils down to `b[c:c+1] += j`, an in-place add-update of one element of the gradient ref, with no dense one-hot array. How does this work? During transposition, each primitive's backward rule is handed a *gradient accumulator* for each input, and when a gradient ref is bound the accumulator is backed by that ref. Built-in primitives like `dynamic_slice` (which implements the indexing above) check for a ref-backed accumulator and perform an in-place add-update of just the entries they touched. If you define custom derivatives with hijax, you can make your own primitives do the same: implement the general `vjp_bwd` method (rather than the convenience wrapper `vjp_bwd_retval`) and handle the `RefAccum` case with an in-place update. See {ref}`jax-301-vjp-bwd-accums` for details and a worked example. ## `DontWant`: skipping unneeded gradients The VJP function returned by `jax.vjp` computes gradients for all the arguments of the differentiated function. But sometimes we don't need all of them, like when differentiating with respect to parameters but not data. Passing `jax.ad.DontWant()` for an argument tells the backward pass not to compute its gradient at all, playing the same role for `jax.vjp` that `argnums` plays for `jax.grad`: ```{code-cell} def predict(W, x): return W @ x W = jnp.ones((4, 4)) x = jnp.ones(4) _, f_vjp = jax.vjp(predict, W, x) W_bar, x_bar = f_vjp.with_refs(jax.ad.GradValue(), jax.ad.DontWant())(jnp.ones(4)) print(W_bar[0]) # [1., 1., 1., 1.] print(x_bar) # DidntWant() ``` This can save real work in the backward pass, like an eager form of dead code elimination. Transpose rules can check for `DontWant` and skip computing the corresponding cotangents. For example, the transpose of matrix multiplication usually computes two dot products, one for each operand's gradient, but with `DontWant` it computes only one. We can see that by counting the `dot_general` operations in the jaxpr of each VJP application: ```{code-cell} _, f_vjp = jax.vjp(predict, W, x) both = jax.make_jaxpr(lambda: f_vjp(jnp.ones(4)))() only_W = jax.make_jaxpr( lambda: f_vjp.with_refs(jax.ad.GradValue(), jax.ad.DontWant())(jnp.ones(4)))() print(str(both).count('dot_general')) # 2, one dot for each gradient print(str(only_W).count('dot_general')) # 1, the dot for x_bar is skipped ``` ## Example: gradient accumulation over microbatches Here's a more realistic recipe that puts these pieces together. In pipelined training, we often split a batch into microbatches, run a forward and backward pass one microbatch at a time, and accumulate weight gradients as we go. Using `with_refs`, each microbatch's backward pass accumulates directly into a single fixed gradient buffer: ```{code-cell} NUM_LAYERS = 3 NUM_MUBATCHES = 5 MUBATCH_SIZE = 7 def mubatch_loss(Ws, xs): # inner loop over layers act, _ = jax.lax.scan(lambda x, W: (jnp.dot(x, W), None), xs, Ws) return jnp.mean(act) def process_batch(Ws, xs_batch): grad_acc = jax.new_ref(jnp.zeros_like(Ws)) def process_mubatch(_, xs): loss, f_vjp = jax.vjp(lambda Ws: mubatch_loss(Ws, xs), Ws) f_vjp.with_refs(grad_acc)(jnp.ones_like(loss)) # accumulate in-place return (), loss xs_mubatches = xs_batch.reshape(NUM_MUBATCHES, MUBATCH_SIZE, -1) # outer loop over microbatches (), losses = jax.lax.scan(process_mubatch, (), xs_mubatches) return jax.freeze(grad_acc), losses Ws = jnp.ones((NUM_LAYERS, 4, 4)) xs_batch = jnp.ones((NUM_MUBATCHES * MUBATCH_SIZE, 4)) grads, losses = process_batch(Ws, xs_batch) ``` Each iteration of the outer scan runs a forward and backward pass for one microbatch, and `with_refs(grad_acc)` makes the backward pass add that microbatch's gradient contribution directly into `grad_acc`. Note that the scan body closes over `grad_acc`, which is fine for `scan` (though it wouldn't be for `vmap` or `shard_map`; see the ref restrictions in {ref}`jax-101-refs`). Once all the microbatches are processed, we `freeze` the accumulator to get the total batch gradient as an immutable `Array`. The result matches what we'd get by differentiating the summed loss directly: ```{code-cell} xs_mubatches = xs_batch.reshape(NUM_MUBATCHES, MUBATCH_SIZE, -1) grads_expected = jax.grad( lambda Ws: sum(mubatch_loss(Ws, xs) for xs in xs_mubatches))(Ws) print(jnp.allclose(grads, grads_expected, atol=1e-3, rtol=1e-3)) # True ``` But unlike that version, the ref-based version never materializes a whole-model gradient per microbatch: each layer's contribution is added straight into one gradient buffer, allocated once, no matter how many microbatches we process. ## `foreach`, a new way to write `scan` `jax.lax.scan` is a loop construct with a built-in fixed access pattern for scanned-over inputs and outputs. The access pattern is built in for autodiff reasons: if we were instead to slice into immutable inputs directly, reverse-mode autodiff would end up creating one-hot gradients and summing them up, as in the `take` example above, which can be asymptotically inefficient. See [Sec 5.3.3 of the Dex paper](https://arxiv.org/pdf/2104.05372). But reading slices of `Ref`s doesn't have this efficiency problem: under reverse-mode autodiff, those reads always become in-place accumulation operations. As a result, we no longer need to be constrained by `scan`'s fixed access pattern. We can write more flexible loops, e.g. with non-sequential access. Moreover, having mutation available allows for some syntax tricks, like in this recipe for a `foreach` decorator: ```{code-cell} import jax import jax.numpy as jnp from jax.lax import scan def foreach(*args): def decorator(body): return scan(lambda _, elts: (None, body(*elts)), None, args)[1] return decorator ``` ```{code-cell} r = jax.new_ref(0) xs = jnp.arange(10) @foreach(xs) def ys(x): r[...] += x return x * 2 print(r) # Ref(45, dtype=int32) print(ys) # [ 0 2 4 6 8 10 12 14 16 18] ``` Here, the loop runs immediately, updating `r` in place and binding `ys` to the stacked outputs of the body. ### Reading more than one element per iteration Looping over indices, rather than over the data itself, lets each iteration read whatever it needs from a ref. A common case that's awkward with `scan`'s access pattern is a loop that needs both `x[n]` and `x[n-1]`: with `scan`, you'd scan over two shifted copies of the input, `(xs[1:], xs[:-1])`, or thread the previous element through the loop carry. With a ref, each iteration just reads both. Here's a first-order high-pass filter, $y_n = \alpha (y_{n-1} + x_n - x_{n-1})$, with its state in a ref too: ```{code-cell} def highpass(xs, alpha=0.9): xs_ref = jax.new_ref(xs) y_ref = jax.new_ref(0.) @foreach(jnp.arange(1, len(xs))) def ys(n): y_ref[...] = alpha * (y_ref[...] + xs_ref[n] - xs_ref[n - 1]) return y_ref[...] return ys ``` It matches a `scan` version that zips shifted copies, gradients included: ```{code-cell} def highpass_scan(xs, alpha=0.9): def step(y, x_and_prev): x, x_prev = x_and_prev y = alpha * (y + x - x_prev) return y, y return scan(step, 0., (xs[1:], xs[:-1]))[1] xs = jnp.sin(jnp.arange(8.)) + jnp.arange(8.) / 4 loss = lambda f: lambda xs: jnp.sum(f(xs) ** 2) print(jnp.allclose(highpass(xs), highpass_scan(xs))) print(jnp.allclose(jax.grad(loss(highpass))(xs), jax.grad(loss(highpass_scan))(xs))) ``` And the backward pass is what we'd hope for: ```{code-cell} ys, highpass_vjp = jax.vjp(highpass, xs) print(jax.make_jaxpr(highpass_vjp)(jnp.ones_like(ys))) ``` Each iteration of the reversed scan adds into exactly two entries of a single `f32[8]` gradient ref, `h[j]` and `h[k]`, one for `x[n]` and one for `x[n-1]`, with no shifted copies of the input and no one-hot arrays. The same approach works for any fixed set of neighbors, like reading `x[n-2]`, `x[n-1]`, and `x[n]` together.